How to create and train (with mutation and selection) a neural network to predict the next state of a board?
Re: Should I also have it randomly delete nodes and connections, in case they also make improvements?
You may read about "dropout". In this case it's not preferred since overfitting is actually a benefit.  

Re: Should the nodes in the hidden layer perform a logical AND on all of their inputs or an OR?
There are no logical operations in Neural Networks. You need to multiply the input by the weights. At the end you will arrive to a fraction number that you need to convert back to 1 or 0, here I suggest to look into "softmax" that is an activation function exactly for this.  

Re:how do I know if it will ever get above 90% accuracy, or even just 50%?
As Jaden suggested you will need a benchmark. You may simply take my results from your other post. (i.e 100% in 20 epochs)  

I agree with Jaden, you have a lot on your table. Perhaps it would make sense to first develop a DNN to learn the GOL, you will need the activations, the loss calculation and at least one optimizer. Once this is tested and working you could start with the second model for the genetic algorithm that will need a different loss function and also the genetic operations, population selection etc. Nice big project, good luck!